↑

Properties

Label 7.7.78373945.1
Degree $7$
Signature $[7, 0]$
Discriminant $78373945$
Root discriminant \(13.42\)
Ramified primes $5,13,1205753$
Class number $1$
Class group trivial
Galois group $S_7$ (as 7T7)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1)
 
Copy content gp:K = bnfinit(y^7 - y^6 - 7*y^5 + 3*y^4 + 13*y^3 + y^2 - 4*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1)
 

\( x^{7} - x^{6} - 7x^{5} + 3x^{4} + 13x^{3} + x^{2} - 4x - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $7$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[7, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(78373945\) \(\medspace = 5\cdot 13\cdot 1205753\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(13.42\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}13^{1/2}1205753^{1/2}\approx 8852.906020059176$
Ramified primes:   \(5\), \(13\), \(1205753\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{78373945}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $6$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a$, $a^{6}-a^{5}-7a^{4}+3a^{3}+12a^{2}+a-1$, $a^{6}-a^{5}-7a^{4}+3a^{3}+13a^{2}+a-3$, $a^{6}-a^{5}-7a^{4}+3a^{3}+13a^{2}-3$, $4a^{6}-6a^{5}-25a^{4}+24a^{3}+40a^{2}-14a-9$, $2a^{6}-3a^{5}-12a^{4}+11a^{3}+18a^{2}-4a-4$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 34.6446938558 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 34.6446938558 \cdot 1}{2\cdot\sqrt{78373945}}\cr\approx \mathstrut & 0.250455658486 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^7 - x^6 - 7*x^5 + 3*x^4 + 13*x^3 + x^2 - 4*x - 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_7$ (as 7T7):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for $S_7$
Character table for $S_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: deg 14
Degree 21 sibling: deg 21
Degree 30 sibling: deg 30
Degree 35 sibling: deg 35
Degree 42 siblings: deg 42, deg 42, deg 42, deg 42
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.7.0.1}{7} }$ R ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.7.0.1}{7} }$ R ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.7.0.1}{7} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.5.1.0a1.1$x^{5} + 4 x + 3$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(1205753\) Copy content Toggle raw display $\Q_{1205753}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*5040 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.78373945.2t1.a.a$1$ $ 5 \cdot 13 \cdot 1205753 $ \(\Q(\sqrt{78373945}) \) $C_2$ (as 2T1) $1$ $1$
6.295...625.14t46.a.a$6$ $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $6$
*5040 6.78373945.7t7.a.a$6$ $ 5 \cdot 13 \cdot 1205753 $ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $6$
14.377...625.21t38.a.a$14$ $ 5^{4} \cdot 13^{4} \cdot 1205753^{4}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $14$
14.874...625.42t413.a.a$14$ $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $14$
14.111...625.30t565.a.a$14$ $ 5^{9} \cdot 13^{9} \cdot 1205753^{9}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $14$
14.295...625.30t565.a.a$14$ $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $14$
15.295...625.42t412.a.a$15$ $ 5^{5} \cdot 13^{5} \cdot 1205753^{5}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $15$
15.874...625.42t411.a.a$15$ $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $15$
20.874...625.70.a.a$20$ $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $20$
21.874...625.84.a.a$21$ $ 5^{10} \cdot 13^{10} \cdot 1205753^{10}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $21$
21.685...625.42t418.a.a$21$ $ 5^{11} \cdot 13^{11} \cdot 1205753^{11}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $21$
35.764...625.126.a.a$35$ $ 5^{20} \cdot 13^{20} \cdot 1205753^{20}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $35$
35.258...625.70.a.a$35$ $ 5^{15} \cdot 13^{15} \cdot 1205753^{15}$ 7.7.78373945.1 $S_7$ (as 7T7) $1$ $35$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)